revenue optimization in the generalized second-price auctionkevinlb/talks/2013-gsp-revenue.pdf ·...

Model and Auctions Theory: single slot All pure equilibria, perfect information Equilibrium selection

Revenue Optimization in theGeneralized Second-Price Auction

Kevin Leyton-Brown

joint work with David R. M. ThompsonTo appear at EC’13

{daveth, kevinlb}@cs.ubc.ca, University of British Columbia

April 18, 2013; Dagstuhl

Revenue Optimization in the GSP Leyton-Brown (joint work with David Thompson)


Introduction

Despite years of research into novel designs, search engines haveheld on to (quality-weighted) GSP.

Question

How can revenue be maximized within the GSP framework?

Various (reserve price; squashing) schemes have been proposed.

We do three kinds of analysis:

theoretical: single slot, Bayesian

computational, perfect information: enumerate all pureequilibria; consider best and worst

computational: consider the equilibrium corresponding to aDS truthful mechanism with the appropriate allocation rule



Outline

1 Model and auctions

2 Theoretical analysis, single-slot auctions

3 What happens in the multi-slot case?

4 Equilibria corresponding to DS truthful mechanisms



Modeling advertisers

Definition (Varian’s model [Varian 07])

Each advertiser i has a valuation vi per click, and quality score qi.In position k, i’s ad will be clicked with probability αkqi, where αk

is a position-specific click factor.



“Vanilla” GSP

rank by biqi, charge lowest amount that would preserveposition in the ranking.

1 slot, 2 bidders, quality scores q1 = 1 and q2 = 0.5:

0.2 0.4 0.6 0.8 1.0v1

0.2

0.4

0.6

0.8

1.0v 2



GSP with Squashing

rank by bi(qi)s, s ∈ [0, 1] [Lahaie, Pennock 07].

s = 1: vanilla GSPs = 0: no quality weighting

used in practice by Yahoo!, according to media reports

0.2 0.4 0.6 0.8 1.0v1

0.2

0.4

0.6

0.8

1.0v 2

1 slot, 2 bidders, quality scores q1 = 1 and q2 = 0.5, s = 0.19.



GSP with unweighted reserves (UWR)

Vanilla GSP with global minimum bid and payment of r

UWR was common industry practice; now replaced by QWR.

0.2 0.4 0.6 0.8 1.0v1

0.2

0.4

0.6

0.8

1.0

v 2

0.2 0.4 0.6 0.8 1.0v1

0.2

0.4

0.6

0.8

1.0

v 2

1 slot, 2 bidders, quality scores q1 = 1 and q2 = 0.5, r = .549.



GSP with quality-weighted reserves (QWR)

Vanilla GSP with per-bidder minimum bid and payment r/qiUWR was common industry practice; now replaced by QWR.

0.2 0.4 0.6 0.8 1.0v1

0.2

0.4

0.6

0.8

1.0

v 2

1 slot, 2 bidders, quality scores q1 = 1 and q2 = 0.5, r = .375.



GSP with unweighted reserves and squashing (UWR+sq)

0.2 0.4 0.6 0.8 1.0v1

0.2

0.4

0.6

0.8

1.0

v 2

1 slot, 2 bidders, quality scores q1 = 1 andq2 = 0.5, r = .505, s = 0.32.



GSP: quality-weighted reserves and squashing (UWR+sq)

0.2 0.4 0.6 0.8 1.0v1

0.2

0.4

0.6

0.8

1.0

v 2

1 slot, 2 bidders, quality scores q1 = 1 andq2 = 0.5, r = .472, s = 0.24.



Our main findings

Considering Varian’s valuation model, our main findings:

QWR is consistently the lowest-revenue reserve-price variant,and substantially worse than UWR.

Anchoring: a new GSP variant that is provably optimal insome settings, and does well in others

first systematic investigation of the interaction betweenreserve prices and squashing

first systematic investigation of the effect of equilibriumselection on the effectiveness of revenue optimization



Outline







Revenue-optimal position auctions

The auctioneer is selling impressions. A bidder’sper-impression valuation is qivi, where:

the auctioneer knows qithe auctioneer knows the distribution from which vi comes

Thus, even if per-click valuations are i.i.d., each bidder has adifferent per-impression valuation distribution, and the sellerknows about those differences.

Strategically, it doesn’t matter how q’s are distributed, becauseit is impossible for a bidder to participate in the auctionwithout revealing this information.



Optimality of unweighted reserves

PropositionConsider any one-position setting where each agent i’s per-click valuation vi isindependently drawn from a common distribution g. If g is regular, then theoptimal auction uses the same per-click reserve price r for all bidders.

Proof.

Because g is regular, we must maximize virtual surplus.

i’s value per-impression is qivi.

Transforming g into a per-impression valuation distribution f gives:f(qivi) = g(vi)/qi and F (givi) = G(vi).

Substituting into the virtual value function gives:

ψi(qivi) = qi

(vi −

1−Gi(vi)

gi(vi)

)Optimal per-click reserve ri is solution to ψi(qiri) = 0, which isindependent of qi. 2



Uniform distribution, single slot

Definition (Anchoring GSP)

Bidders face an unweightedreserve r, and those who exceedit are ranked by (bi − r)qi.

Proposition

When per-click valuations aredrawn from the uniformdistribution, anchoring GSP isoptimal. 0.2 0.4 0.6 0.8 1.0

v1

0.2

0.4

0.6

0.8

1.0

v 2



Optimizing GSP variants by grid search: uniform, 2 bidders

Auction Revenue (±1e− 5) Parameters

VCG/GSP 0.208 —Squashing 0.255 s = 0.19

QWR 0.279 r = 0.375UWR 0.316 r = 0.549

QWR+Sq 0.321 r = 0.472, s = 0.24UWR+Sq 0.322 r = 0.505, s = 0.32Anchoring 0.323 r = 0.5

Anchoring’s r agrees with [Myerson 81] and QWR’s with [Sun,

Zhou, Deng 11].

Optimal parameters for other variants don’t correspond torecommendations from the literature.



Optimal auction for the log-normal distribution

Anchoring is not always optimal(but perhaps it is always a good approximation?)

0.5 1.0 1.5 2.0 2.5v1

0.5

1.0

1.5

2.0

2.5

v 2

Optimal auction for log normal, 1 slot, 2 bidders, quality scoresq1 = 1 and q2 = 0.5. Anchoring shown for comparison.



Outline







Computing equilibria

Action-graph games (AGGs) exploit structure to representgames in exponentially less space than than the normal form[Bhat, LB 04; Jiang, LB 06; Jiang, LB, Bhat 11].

Games involving GSP and Varian’s preference model havesuch structure [Thompson, LB 09].

Heuristic tree search can enumerate all pure-strategy Nashequilibria of an AGG [Thompson, Leung, LB 11].

S1={H} S1={T} S1={H,T}

S2={H} S2={T} S2={H,T}



Investigating multiple slots with grid search

Leverage AGGs to consider more than a single slot, and toexamine different equilibria of GSP variants to determineimpact of equilibrium selection

Sample perfect-information games from the distribution overvalues and quality scores

5 bidders; 26 bid increments each; 3 slots; uniform valuations

enumerate pure-strategy equilibriaconsider statistics over their best and worst (conservative) NE.

Identify optimal parameter settings by performing fine-grainedgrid search.



Equilibrium Selection and Reserve Prices

0 5 10 15 20 25r

0

2

4

6

8

10

12

14

Revenue

Worst NE

Best NE

UWR

0 5 10 15 20 25r

0

2

4

6

8

10

12

14

Revenue

Worst NE

Best NE

QWR

0 5 10 15 20 25r

0

2

4

6

8

10

12

14

Revenue

Worst NE

Best NE

Anchoring

Any reserve scheme dramatically improves vanilla GSP’sworst-case revenue (look at reserves of $0).

Optimal unweighted reserves are higher than quality-weighted.

High bidding can do the work of high reserve prices. Thus,worst-case reserve prices tend to be higher than best case.



Equilibrium Selection and Squashing

0.0 0.2 0.4 0.6 0.8 1.0s

0

2

4

6

8

10

12

14

Revenue

Worst NE

Best NE

Squashing can improve revenue in best- and worst-caseequilibrium. (Recall: s = 1 is vanilla GSP.)

Smaller impact, lower sensitivity than reserve prices.

Gap between best and worst is consistently large (∼ 2.5×).



Comparing variants optimized for best/worst case

Auction Revenue

Vanilla GSP 3.814Squashing 4.247

QWR 9.369Anchoring 10.212QWR+Sq 10.217UWR 11.024

UWR+Sq 11.032

Worst-case equilibrium

Auction Revenue

Vanilla GSP 9.911QWR 10.820

Squashing 11.534UWR 11.686

Anchoring 12.464QWR+Sq 12.627UWR+Sq 12.745

Best-case equilibrium

Worst case: 2-way tie (UWR+Sq, UWR)

Best case: 3-way tie (UWR+Sq, QWR+Sq, Anchoring)

UWR’s worst case is better than QWR’s best case.



Outline







Equilibrium Selection

With vanilla GSP, it’s common to study the equilibrium that leadsto the efficient (thus, VCG) outcome. Many reasons why this is aninteresting equilibrium:

Existence, uniqueness, polytime computability [Aggarwal et al 06]

Envy-free, symmetric, competitive eq [Varian 07; EOS 07]

Impersonation-proof [Kash, Parkes 12]

Doesn’t predict that GSP gets more revenue than Myerson(“Non-contradiction criterion”) [ES 10]

Analogously, can compute the equilibrium corresponding to a DStruthful mechanism with the appropriate allocation rule.

see previous analyses of squashing [LP 07] and reserves [ES 10].



Distributions

For these experiments, we used two distributions:

Uniform vi’s drawn from uniform (0, 25); qi’s drawn fromuniform (0, 1).

Log-Normal qi’s and vi’s drawn from log-normaldistributions; qi positively correlated with vi by Gaussiancopula. (Similar to [LP07]; new parameters based on personalcommunication.)

We compute equilibrium following recursion of [Aggarwal et al 06].We optimize parameters by grid search.



Revenue across GSP variants, optimal parameters

Auction Revenue

VCG 7.737Squashing 9.123

QWR 10.598UWR 12.026

QWR+Sq 12.046Anchoring 12.2UWR+Sq 12.220

Uniform distribution

Auction Revenue

VCG 20.454QWR 48.071

Squashing 53.349QWR+Sq 79.208

UWR 80.050Anchoring 80.156UWR+Sq 81.098

Log-Normal Distribution



Reserve Prices

0 5 10 15 20 25Reserve Price

0

2

4

6

8

10

12

14

Revenue

anchor

uRes

wRes

Uniform Distribution

100 101 102 103 104 105

Reserve Price

0

10

20

30

40

50

60

70

80

90

Revenue

anchor

uRes

wRes


All three reserve-based variants (anchoring, QRW and UWR)provide substantial revenue gains (compare to reserve 0).

Anchoring very slightly better than UWR; both substantiallybetter than QWR.



Squashing + UWR


0

2

4

6

8

10

12

14

Revenue

0.0

0.25

0.5

0.75

1.0


100 101 102 103 104 105

Reserve Price

0

10

20

30

40

50

60

70

80

90

Revenue

0.0

0.25

0.5

0.75

1.0


Adding squashing to UWR provides small marginalimprovements (compare to s = 1) and does not substantiallyaffect the optimal reserve price.



Squashing + QWR


0

2

4

6

8

10

12

14

Revenue

0.0

0.25

0.5

0.75

1.0


100 101 102 103 104 105

Reserve Price

0

10

20

30

40

50

60

70

80

90

Revenue

0.0

0.25

0.5

0.75

1.0


Adding squashing to QWR yields big improvements (compareto s = 1); high sensitivity.But, the higher the squashing power (s→ 0), the less reserveprices are actually weighted by quality.Log-normal: optimal parameter setting (s = 0.0) removesquality scores entirely and is thus equivalent to UWR.



Does squashing help QWR via reserve or ranking?

100 101 102 103 104 105

Reserve Price

0

10

20

30

40

50

60

70

80

90R

evenue

0.0

0.25

0.5

0.75

1.0

Squashing applied to reserve only(log normal)

100 101 102 103 104 105

Reserve Price

0

10

20

30

40

50

60

70

80

90

Revenue

0.0

0.25

0.5

0.75

1.0

Squashing applied to ranking only(log normal)

Applying squashing only to reserve prices can dramaticallyincrease QWR’s revenue (compare to s = 1).

However, there has to be a lot of squashing (i.e., s close to 0)optimal reserve is very dependent on squashing poweroptimal parameter setting is s = 0: identical to UWR

Applying squashing only to ranking, the marginal gains fromsquashing over QWR (with optimal reserve) are very small.



Scaling

Because equilibrium computation is cheap, we can scale up.

2 4 6 8 10 12 14 16 18 20Agents

0

5

10

15

20

25

Revenue

VCG

Squashing

UWR

QWR

Anchoring

UWR+Sq

QWR+Sq


2 4 6 8 10 12 14 16 18 20Agents

0

50

100

150

200

250

300

350

Revenue

VCG

Squashing

UWR

QWR

Anchoring

UWR+Sq

QWR+Sq


Top 4 mechanisms are still nearly tied. Squashing and QWRare consistently below.

As n increases, squashing gains on QWR.

For log normal, squashing substantially outperforms QWR.



Conclusions

We optimized revenue in GSP-like auctions under Varian’svaluation model, conducting three different kinds of analysis.

QWR was consistently the lowest-revenue reserve-pricevariant, and substantially worse than UWR.

Anchoring does well; optimal in simple settings

Equilibrium selection: vanilla GSP, squashing have big gapsbetween best and worst case

Squashing helps both UWR and QWR.

Why do search engines prefer QWR to UWR? Possible explanations:

Whoops—they should use UWR.

Analysis should consider long-run revenue

Analysis should consider cost of showing bad ads

Actually, they do some other, secret thing, not QWR.


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